Lecture 12. Asset pricing model. Randall Romero Aguilar, PhD I Semestre 2017 Last updated: June 15, 2017

Transcription

1 Lecture 12 Asset pricing model Randall Romero Aguilar, PhD I Semestre 2017 Last updated: June 15, 2017 Universidad de Costa Rica EC Teoría Macroeconómica 2

2 Table of contents 1. Introduction 2. The model 3. Consumption based capital asset pricing model

3 Introduction

4 Objective Asset pricing theory tries to explain why some assets pay higher average returns than others. Accordingly, the objective is to understand the prices or values of claims to uncertain payments. 1

5 Risk versus return The central aspect is the risk-return tradeoff. It is rational that investors demand additional return for an asset incorporating more risk. Problems arise, however, when one tries to determine the relevant risk factors and their expected compensation. 2

6 Theory beginnings The basis for this theory was already laid in the 1950s and 60s with the portfolio selection theory by Markowitz and the Capital Asset Pricing Model (CAPM) by Sharpe, for which he received a Nobel Prize in The CAPM significantly shaped and changed financial management. 3

7 Current relevance Today it is still widely used in practice and plays the centerpiece in the theoretical discussion of asset pricing, although it continues to be sharply criticized. This leads to a variety of adaptations and further developments of the CAPM, but so far no model has been able to sufficiently persuade financial scientists and practitioners. In this lecture, we derive one of such adaptations: the consumption-based capital asset pricing model. 4

8 The model

9 The model In this lecture we consider an endowment economy where there are many assets. The model we develop is similar to the Consumption and financial assets model that we studied in lecture 8. In addition to having a single risk-free asset, we now assume that there are N risky assets 5

10 Future dividends For asset i {0, 1,..., N} and time t we have: y it dividend of asset P it price of asset (ex-dividend) S it number of shares Notice that the asset is priced after it has paid dividends to its current owner. 6

11 Stochastic process of dividends Define the vector y t {y 0t, y 1t,..., y Nt } Let x t be an informative vector variable that helps predicts future dividends. Let z t (y t, x t ). We assume that z t follows a Markov process: F t (z t+1 ) = F (z t+1 z t ) 7

12 The consumer Infinite horizon Instant utility u(c t ) depends on current consumption Constant utility discount rate β (0, 1) Lifetime expected utility is: [ ] E β s t u (c s ) z t s=t 8

13 Budget constrain In this model, the consumer is endowed with {s 0t, s 1t,..., s Nt } shares. Each share of asset i entitles the consumer to receive y it in dividends, and can be sold for P it. Then, the total liquidity available to the consumer at t is N a t (y it + P it ) s it i=0 Consumer uses liquidity a t to pay for consumption c t and to buy new shares s it+1 : N c t + P it s it+1 i=0 9

14 The consumer problem The consumer problem consist of finding: a consumption rule c t = c(a t, z t ) a portfolio rule s i,t+1 = s i (a t, z t ) i = {0, 1,..., N} to maximize expected utility: [ ] max E β s t u (c s ) z t c t,s 0:N,t s=t c t + subject to N P it s it+1 = i=0 N (y it + P it ) s it a t i=0 10

15 The Bellman equation V (a t, z t ) = max {u(c t ) + β E [V (a t+1, z t+1 ) z t ]} Use the budget constraint to substitute c t and the definition of liquidity to substitute a t+1 : V (a t, z t ) = max s i,t+1 [ β E { u ( a t ) N P it s it i=0 ( N ) V (y i,t+1 + P i,t+1 ) s i,t+1, z t+1 i=0 z t ]} 11

16 Solving the consumer problem Taking derivative with respect to s it+1 : [ ] V P it u (at+1, z t+1 ) (c t ) + β E (y i,t+1 + P i,t+1 ) z t = 0 a t+1 The envelope condition is Thus, the Euler equation is: V (a t, z t ) a t = u (c t ) P it u (c t ) = β E [ u (c t+1 ) (y i,t+1 + P i,t+1 ) z t ] 12

17 Side note: Conditional expectation The expression E t [X t ] E [X t+1 z t ] denotes the expected value of X t+1 based on information z t up to time t. 13

18 Side note: The law of iterated expectations For two arbitrary random variables y and z, the law of iterated expectations says that E(y) = E [E(y z)] In words, the unconditional expectation of the conditional expectation of y conditional on z is equal to the unconditional expectation of y. This has the following implication for a time series: E t [E t+1 (x t+2 )] = E t (x t+2 ) In other words, your current best guess of your best guess next period of the realization of x two periods from now is equal to your current best guess of x two periods from now. 14

19 Example 1: A risk neutral consumer

20 Assume that consumer is risk neutral: u(c) = c u (c) = 1 Then the price of a share of asset i is P it = β E [y i,t+1 + P i,t+1 z t ] or equivalently P it = β E t [y i,t+1 + P i,t+1 ] Notice that this implies that P it+1 = β E t+1 [y i,t+2 + P i,t+2 ] 15

21 Substitution of the P it+1 of the equation for P it results in: P it = β E t [y i,t+1 + P i,t+1 ] = β E t [y i,t+1 + β E t+1 [y i,t+2 + P i,t+2 ]] = β E t [y i,t+1 ] + β 2 E t [E t+1 [y i,t+2 + P i,t+2 ]] = β E t [y i,t+1 ] + β 2 E t [y i,t+2 + P i,t+2 ] Last step follows from the Law of Iterated Expectations. If we keep substituting future prices we get: P it = E t [ s=t+1 β s t y is ] + lim s E t (β s P s ) As long as there is no bubles, the limit term is zero. 16

22 As long as there is no bubles, the limit term is zero. [ ] P it = E t β s t y is s=t+1 This says that the price of one share of asset i must be equal to the expected discounted value of all future dividends. 17

23 The Euler equation Define the (random) return on asset i by R it y it+1 + P it+1 P it Then, we can write the Euler equation as: u (c t ) = β E t [ u (c t+1 ) R it ] At time t, c t can be considered non-random. Therefore: 1 = E t β u (c t+1 ) u (c t ) MRS between consumption in t+1 and t R it gross rate of return on asset i E t [M t R it ] 18

24 A riskless asset Asset 0 is riskless (a discount index bond). Its yields are: ( ) t+1 y 0 = 1, t+2 0, t+3 0,... and its future price is P 0,t+1 = 0. So its return is This implies that: R 0t y 0t+1 + P 0t+1 P 0t = 1 P 0t 1 = E t [M t R 0t ] = R 0t E t [M t ] 1 R 0t = E t [M t ] 19

25 A formula for risk premium Remember that Cov[A, B] = E[AB] E[A] E[B] It follows that: 1 = E t [M t R it ] = E t [M t ] E t [R it ] + Cov t [M t, R it ] 1 E t [M t ] = E t [R it ] + Cov t [M t, R it ] = R 0t E t [M t ] risk premium E t [R it ] R 0t = Cov t [M t, R it ] risk premium of asset i E t [M t ] 20

26 Consumption based capital asset pricing model

27 Market portfolio Let s normalize the total number of shares for each asset to s i = 1 Total market dividend is y t+1 = N i=0 y it+1 Since this is an endowment economy (no production): c t = y t t 21

28 A claim to aggregate endowment Consider a claim to aggregate endowment next period. ( ) t+1 y M = y t+1, t+2 0, t+3 0,... If its current price is P Mt, then its return is Its risk premium would be: R Mt = y t+1 P Mt E t [R Mt ] R 0t = Cov t [M t, R Mt ] E t [M t ] 22

29 Quadratic utility Assume that u(c) = αc 0.5c 2 In this case M t βu (c t+1 ) u (c t ) = βα α y t βy t+1 α y t u (c) = α c = β (α y t+1) α y t = βα α y t βp Mt α y t R Mt ϕ 1 + ϕ 2 R Mt With quadratic utility, the MRS of future consumption for present consumption is an affine function of the market return. 23

30 Comparing risk premia The ratio of the risk premium of asset i to market risk premium is: E t [R it ] R 0t = Cov t [M t, R it ] E t [R Mt ] R 0t Cov t [M t, R Mt ] = Cov t [ϕ 1 + ϕ 2 R Mt, R it ] Cov t [ϕ 1 + ϕ 2 R Mt, R Mt ] = ϕ 2 Cov t [R Mt, R it ] ϕ 2 Cov t [R Mt, R Mt ] = Cov t [R Mt, R it ] Var t [R Mt ] β it The β it term is a measure of the comovement of the return of asset i with that of the market. 24

31 Consumption-based CAPM From last expression, we get the consumption-based capital asset pricing model: CCAPM E t [R it ] R 0t risk premium of asset i = Cov t [R Mt, R it ] [E t [R Mt ] R 0t ] Var t [R Mt ] market risk premium 25